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Here, we prove the uniform observability of a two-grid method for the semi-discretization of the 1D-wave equation for a time $T>2\sqrt{2}$ ; this time, if the observation is made in $(-T/2,T/2)$ , is optimal and this result improves an earlier work of Negreanu and Zuazua [C. R. Acad. Sci. Paris Sér. I338 (2004) 413–418].Our proof follows an Ingham type approach.

This paper deals with theobservability analysis and the observer synthesis of a class ofnonlinear systems. In the single output case, it is known [4-6] that systems whichare observable independently of the inputs, admit an observablecanonical form. These systems are called uniformly observablesystems. Moreover, a high gain observer for these systems can bedesigned on the basis of this canonical form. In this paper, weextend the above results to multi-output uniformly observablesystems. Corresponding canonical forms are presented andsufficient conditions which permit the design of constant and highgain observers for these systems aregiven.

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An Ingham type proof for a two-grid observability theorem

Nonlinear observers for locally uniformly observable systems

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2 results

An Ingham type proof for a two-grid observability theorem

Nonlinear observers for locally uniformly observable systems